Tying up the Loose Ends : A Mathematically Knotted Protein

Knots have attracted scientists in mathematics, physics, biology, and engineering. Long flexible thin strings easily knot and tangle as experienced in our daily life. Similarly, long polymer chains inevitably tend to get trapped into knots. Little is known about their formation or function in proteins despite >1,000 knotted proteins identified in nature. However, these protein knots are not mathematical knots with their backbone polypeptide chains because of their open termini, and the presence of a "knot" depends on the algorithm used to create path closure. Furthermore, it is generally not possible to control the topology of the unfolded states of proteins, therefore making it challenging to characterize functional and physicochemical properties of knotting in any polymer. Covalently linking the amino and carboxyl termini of the deeply trefoil-knotted YibK from Pseudomonas aeruginosa allowed us to create the truly backbone knotted protein by enzymatic peptide ligation. Moreover, we produced and investigated backbone cyclized YibK without any knotted structure. Thus, we could directly probe the effect of the backbone knot and the decrease in conformational entropy on protein folding. The backbone cyclization did not perturb the native structure and its cofactor binding affinity, but it substantially increased the thermal stability and reduced the aggregation propensity. The enhanced stability of a backbone knotted YibK could be mainly originated from an increased ruggedness of its free energy landscape and the destabilization of the denatured state by backbone cyclization with little contribution from a knot structure. Despite the heterogeneity in the side-chain compositions, the chemically unfolded cyclized YibK exhibited several macroscopic physico-chemical attributes that agree with theoretical predictions derived from polymer physics.Peer reviewe

Random-Coil Behavior of Chemically Denatured Topologically Knotted Proteins Revealed by Small-Angle X‑ray Scattering

Recent studies on the mechanisms by which topologically knotted proteins attain their natively knotted structures have intrigued theoretical and experimental biophysicists. Of particular interest is the finding that YibK and YbeA, two small trefoil knotted proteins, remain topologically knotted in their chemically denatured states. Using small-angle X-ray scattering (SAXS), we examine whether these chemically denatured knotted proteins are different from typical random coils. By revisiting the scaling law of radius of gyration (<i>R</i>_g) as a function of polypeptide chain length for chemically denatured proteins and natively folded proteins, we find that the chemically denatured knotted proteins in fact follow the same random coil-like behavior, suggesting that the formation of topological protein knots do not necessarily require global compaction while the loosely knotted polypeptide chains are capable of maintaining the correct chirality without defined secondary or tertiary structures

Entangled Proteins: Knots, Slipknots, Links, and Lassos

In recent years the studies of entangled proteins have grown into the whole new, interdisciplinary and rapidly developing field of research. Here we present various types of entangled proteins studied within this field, which form knots, slipknots, links, and lassos. We discuss their geometric features and indicate what biological and physical role the entanglement plays. We also discuss mathematical tools necessary to analyze such structures and present databases and servers assembling information about entangled proteins: KnotProt, LinkProt, and LassoProt